Logarithmic good reduction of abelian varieties

TL;DR

This paper establishes a logarithmic criterion for good reduction of abelian varieties over discretely valued fields, extending classical results and providing new models in the logarithmic setting.

Contribution

It proves a logarithmic version of the Néron-Ogg-Shafarevich criterion for cohomologically tame abelian varieties, ensuring the existence of log smooth models.

Findings

Existence of projective, log smooth models for cohomologically tame abelian varieties.

Generalization of Künnemann's result to the logarithmic setting.

Integration of Gabber's theorem with degeneration theory of abelian varieties.

Abstract

Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the N\'eron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the existence of a projective, regular model of $A$, generalizing a result of K\"unnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.